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# Math 753/852 Finite Differencing

The main things to know here are a few formulae for approximating derivatives using finite differences:

#### First derivative df/dx

Given a set of evenly space gridpoints , where , and a function evaluated at the gridpoints , we can approximate the first derivative of at the gridpoints several ways

One-sided finite differencing for the first derivative, rightwards

One-sided finite differencing for the first derivative, leftwards

Centered finite differencing for the first derivative

In practice you use the fraction on the right-hand-side as an approximation of the derivative, knowing that there is an or error in the approximation.

#### Second derivative

To approximate the second derivative , we use the

Centered finite differencing for the second derivative

#### Other formulae

There are many, many other finite-difference formulae, for higher-order derivatives, higher-order accuracy, and different choices of which gridpoint values enter into the formula. As a starting point, see the following